Let [Formula: see text] be a continuous selfmap of a nontrivial compact metric space [Formula: see text]. This paper derives the large deviations theorem in a sequence. It is shown that if [Formula: see text] satisfies the large deviations theorem in a sequence then it is an E-system which implies that [Formula: see text] is syndetically transitive, and that if the pair [Formula: see text] satisfies the large deviations theorem in a sequence and [Formula: see text] is an infinite space, then [Formula: see text] is syndetically sensitive, which implies that [Formula: see text] is ergodically sensitive. Moreover, it is proved that if the pair [Formula: see text] satisfies the large deviations theorem in a syndetic sequence, then [Formula: see text] is an ergodic measure with supp[Formula: see text]. Our main results extend and improve the corresponding results in the literature.
This paper is mainly concerned with distributional chaos and the principal measure of C 0 -semigroups on a Frechet space. New definitions of strong irregular (semi-irregular) vectors are given. It is proved that if C 0 -semigroup T has strong irregular vectors, then T is distributional chaos in a sequence, and the principal measure μ p ( T ) is 1. Moreover, T is distributional chaos equivalent to that operator T t is distributional chaos for every ∀ t > 0 .
<abstract><p>The consistency and implication relation of chaotic properties of $ p $-periodic discrete system and its induced autonomous discrete system are obtained. The chaotic properties discussed involve several types of transitivity and some stronger forms of sensitivity in the sense of Furstenberg families.</p></abstract>
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